Additional functions for common mathematical operations. More...

Functions
template<typename T , typename U >
constexpr U	playrho::Secant (T target, U a1, T s1, U a2, T s2) noexcept
	Secant method. More...

template<typename T >
constexpr T	playrho::Bisect (T a1, T a2) noexcept
	Bisection method. More...

template<typename T >
constexpr bool	playrho::IsOdd (T val) noexcept
	Is-odd. More...

template<class TYPE >
constexpr auto	playrho::Square (TYPE t) noexcept
	Squares the given value.

template<typename T >
auto	playrho::Atan2 (T y, T x)
	Computes the arc-tangent of the given y and x values. More...

template<typename T , typename = std::enable_if_t< IsIterable<T>::value && IsAddable<decltype(*begin(std::declval<T>()))>::value>
auto	playrho::Average (const T &span)
	Computes the average of the given values.

template<typename T >
std::enable_if_t< IsArithmetic< T >::value, T >	playrho::RoundOff (T value, unsigned precision=100000)
	Computes the rounded value of the given value.

Vec2	playrho::RoundOff (Vec2 value, std::uint32_t precision=100000)
	Computes the rounded value of the given value.

template<typename T , std::size_t N>
constexpr Vector< T, N >	playrho::abs (const Vector< T, N > &v) noexcept
	Absolute value function for vectors.

d2::UnitVec	playrho::abs (const d2::UnitVec &v) noexcept
	Gets the absolute value of the given value.

template<typename T >
constexpr std::enable_if_t< std::is_arithmetic< T >::value, bool >	playrho::AlmostZero (T value)
	Gets whether a given value is almost zero. More...

template<typename T >
constexpr std::enable_if_t< std::is_floating_point< T >::value, bool >	playrho::AlmostEqual (T x, T y, int ulp=2)
	Determines whether the given two values are "almost equal".

template<typename T >
auto	playrho::ModuloViaFmod (T dividend, T divisor) noexcept
	Modulo operation using `std::fmod`. More...

template<typename T >
auto	playrho::ModuloViaTrunc (T dividend, T divisor) noexcept
	Modulo operation using `std::trunc`. More...

Angle	playrho::GetNormalized (Angle value) noexcept
	Gets the "normalized" value of the given angle. More...

template<class T >
Angle	playrho::GetAngle (const Vector2< T > value)
	Gets the angle. More...

template<typename T >
constexpr auto	playrho::GetMagnitudeSquared (T value) noexcept
	Gets the square of the magnitude of the given iterable value. More...

template<typename T >
auto	playrho::GetMagnitude (T value)
	Gets the magnitude of the given value. More...

template<typename T1 , typename T2 >
constexpr auto	playrho::Dot (const T1 a, const T2 b) noexcept
	Performs the dot product on two vectors (A and B). More...

template<class T1 , class T2 , std::enable_if_t< std::tuple_size< T1 >::value==2 &&std::tuple_size< T2 >::value==2, int > = 0>
constexpr auto	playrho::Cross (T1 a, T2 b) noexcept
	Performs the 2-element analog of the cross product of two vectors. More...

template<typename T , typename U >
constexpr auto	playrho::Solve (const Matrix22< U > mat, const Vector2< T > b) noexcept
	Solves A * x = b, where b is a column vector. More...

template<class IN_TYPE >
constexpr auto	playrho::Invert (const Matrix22< IN_TYPE > value) noexcept
	Inverts the given value.

constexpr Vec3	playrho::Solve33 (const Mat33 &mat, const Vec3 b) noexcept
	Solves A * x = b, where b is a column vector. More...

template<typename T >
constexpr T	playrho::Solve22 (const Mat33 &mat, const T b) noexcept
	Solves A * x = b, where b is a column vector. More...

constexpr Mat33	playrho::GetInverse22 (const Mat33 &value) noexcept
	Gets the inverse of the given matrix as a 2-by-2. More...

constexpr Mat33	playrho::GetSymInverse33 (const Mat33 &value) noexcept
	Gets the symmetric inverse of this matrix as a 3-by-3. More...

template<class T >
constexpr auto	playrho::GetRevPerpendicular (const T vector) noexcept
	Gets a vector counter-clockwise (reverse-clockwise) perpendicular to the given vector. More...

template<class T >
constexpr auto	playrho::GetFwdPerpendicular (const T vector) noexcept
	Gets a vector clockwise (forward-clockwise) perpendicular to the given vector. More...

template<std::size_t M, typename T1 , std::size_t N, typename T2 >
constexpr auto	playrho::Transform (const Vector< T1, M > v, const Matrix< T2, M, N > &m) noexcept
	Multiplies an M-element vector by an M-by-N matrix. More...

constexpr Vec2	playrho::Transform (const Vec2 v, const Mat33 &A) noexcept
	Multiplies a vector by a matrix.

constexpr Vec2	playrho::InverseTransform (const Vec2 v, const Mat22 &A) noexcept
	Multiply a matrix transpose times a vector. If a rotation matrix is provided, then this transforms the vector from one frame to another (inverse transform).

constexpr Mat22	playrho::MulT (const Mat22 &A, const Mat22 &B) noexcept
	Computes A^T * B.

Mat22	playrho::abs (const Mat22 &A)
	Gets the absolute value of the given value.

template<typename T >
constexpr T	playrho::NextPowerOfTwo (T x)
	Gets the next largest power of 2. More...

Real	playrho::Normalize (Vec2 &vector)
	Converts the given vector into a unit vector and returns its original length.

Length2	playrho::ComputeCentroid (const Span< const Length2 > &vertices)
	Computes the centroid of a counter-clockwise array of 3 or more vertices. More...

template<typename T >
constexpr T	playrho::GetModuloNext (T value, T count) noexcept
	Gets the modulo next value.

template<typename T >
constexpr T	playrho::GetModuloPrev (T value, T count) noexcept
	Gets the modulo previous value.

Angle	playrho::GetDelta (Angle a1, Angle a2) noexcept
	Gets the shortest angular distance to go from angle 1 to angle 2. More...

constexpr Angle	playrho::GetRevRotationalAngle (Angle a1, Angle a2) noexcept
	Gets the reverse (counter) clockwise rotational angle to go from angle 1 to angle 2. More...

std::vector< Length2 >	playrho::GetCircleVertices (Length radius, unsigned slices, Angle start=0_deg, Real turns=Real{1})
	Gets the vertices for a circle described by the given parameters.

NonNegative< Area >	playrho::GetAreaOfCircle (Length radius)
	Gets the area of a circle.

NonNegative< Area >	playrho::GetAreaOfPolygon (Span< const Length2 > vertices)
	Gets the area of a polygon. More...

SecondMomentOfArea	playrho::GetPolarMoment (Span< const Length2 > vertices)
	Gets the polar moment of the area enclosed by the given vertices. More...

template<typename T , std::size_t N>
constexpr Vector< T, N >	abs (const Vector< T, N > &v) noexcept
	Absolute value function for vectors.

Detailed Description

Additional functions for common mathematical operations.

These are non-member non-friend functions for mathematical operations especially those with mixed input and output types.

Function Documentation

◆ AlmostZero()

template<typename T >

constexpr std::enable_if_t<std::is_arithmetic<T>::value, bool> playrho::AlmostZero ( T value )

constexpr

Gets whether a given value is almost zero.

An almost zero value is "subnormal". Dividing by these values can lead to odd results like a divide by zero trap occurring.

Returns: true if the given value is almost zero, false otherwise.

◆ Atan2()

template<typename T >

auto playrho::Atan2	(	T	y,
		T	x
	)

inline

Computes the arc-tangent of the given y and x values.

Returns: Normalized angle - an angle between -Pi and Pi inclusively.

See also: https://en.cppreference.com/w/cpp/numeric/math/atan2

Referenced by playrho::d2::GetAngle(), and playrho::GetAngle().

◆ Bisect()

template<typename T >

constexpr T playrho::Bisect	(	T	a1,
		T	a2
	)

constexprnoexcept

Bisection method.

See also: https://en.wikipedia.org/wiki/Bisection_method

Referenced by playrho::d2::GetToiViaSat().

◆ ComputeCentroid()

Length2 playrho::ComputeCentroid ( const Span< const Length2 > & vertices )

Computes the centroid of a counter-clockwise array of 3 or more vertices.

Note: Behavior is undefined if there are less than 3 vertices or the vertices don't go counter-clockwise.

Referenced by playrho::d2::PolygonShapeConf::Set().

◆ Cross()

template<class T1 , class T2 , std::enable_if_t< std::tuple_size< T1 >::value==2 &&std::tuple_size< T2 >::value==2, int > = 0>

constexpr auto playrho::Cross	(	T1	a,
		T2	b
	)

constexprnoexcept

Performs the 2-element analog of the cross product of two vectors.

Cross-products the given two values.

Defined as the result of: (a.x * b.y) - (a.y * b.x).

Note: This operation is dimension squashing. I.e. A cross of a 2-D length by a 2-D unit vector results in a 1-D length value.; The unit of the result is the 1-D product of the inputs.; This operation is anti-commutative. I.e. Cross(a, b) == -Cross(b, a).; The result will be 0 if any of the following are true: vector A or vector B has a length of zero; vectors A and B point in the same direction; or vectors A and B point in exactly opposite direction of each other.; The result will be positive if: neither vector A nor B has a length of zero; and vector B is at an angle from vector A of greater than 0 and less than 180 degrees (counter-clockwise from A being a positive angle).; Result will be negative if: neither vector A nor B has a length of zero; and vector B is at an angle from vector A of less than 0 and greater than -180 degrees (clockwise from A being a negative angle).; The absolute value of the result is the area of the parallelogram formed by the vectors A and B.

See also: https://en.wikipedia.org/wiki/Cross_product

Returns: Cross product of the two values.

Note: This operation is anti-commutative. I.e. Cross(a, b) == -Cross(b, a).

See also: https://en.wikipedia.org/wiki/Cross_product

Parameters

a	Value A of a 3-element type.
b	Value B of a 3-element type.

Returns: Cross product of the two values.

Referenced by playrho::d2::ApplyForce(), playrho::d2::ApplyLinearImpulse(), playrho::d2::Simplex::CalcMetric(), playrho::d2::CalcSearchDirection(), playrho::ComputeCentroid(), playrho::d2::Simplex::Get(), playrho::d2::GetConvexHullAsVector(), playrho::d2::GetEffectiveInvMass(), playrho::d2::GetMassData(), playrho::d2::VelocityConstraint::GetPoint(), playrho::GetPolarMoment(), playrho::GetSymInverse33(), playrho::d2::InitVelocity(), playrho::Invert(), playrho::d2::RayCast(), playrho::d2::SetForce(), playrho::Solve(), playrho::Solve33(), playrho::d2::SolvePosition(), playrho::d2::SolveVelocity(), and playrho::d2::Validate().

◆ Dot()

template<typename T1 , typename T2 >

constexpr auto playrho::Dot	(	const T1	a,
		const T2	b
	)

constexprnoexcept

Performs the dot product on two vectors (A and B).

The dot product of two vectors is defined as: the magnitude of vector A, multiplied by, the magnitude of vector B, multiplied by, the cosine of the angle between the two vectors (A and B). Thus the dot product of two vectors is a value ranging between plus and minus the magnitudes of each vector times each other. The middle value of 0 indicates that two vectors are perpendicular to each other (at an angle of +/- 90 degrees from each other).

Note: This operation is commutative. I.e. Dot(a, b) == Dot(b, a).; If A and B are the same vectors, GetMagnitudeSquared(Vec2) returns the same value using effectively one less input parameter.; This is similar to the std::inner_product standard library algorithm except benchmark tests suggest this implementation is faster at least for Vec2 like instances.

See also: https://en.wikipedia.org/wiki/Dot_product

Parameters

a	Vector A.
b	Vector B.

Returns: Dot product of the vectors (0 means the two vectors are perpendicular).

Referenced by playrho::d2::ClipSegmentToLine(), playrho::d2::Simplex::Get(), playrho::d2::GetGearJointConf(), playrho::d2::GetJointTranslation(), playrho::d2::GetLinearVelocity(), playrho::d2::GetManifold(), playrho::d2::VelocityConstraint::GetPoint(), playrho::d2::GetSeparationScenario(), playrho::d2::GetSupportIndex(), playrho::GetSymInverse33(), playrho::d2::InitVelocity(), playrho::InverseTransform(), playrho::MulT(), playrho::d2::RayCast(), playrho::Solve33(), playrho::d2::SolvePosition(), playrho::d2::SolveVelocity(), and playrho::d2::TestPoint().

◆ GetAngle()

template<class T >

Angle playrho::GetAngle ( const Vector2< T > value )

inline

Gets the angle.

Returns: Angular value in the range of -Pi to +Pi radians.

Referenced by playrho::d2::Body::SetAcceleration().

◆ GetAreaOfPolygon()

NonNegative< Area > playrho::GetAreaOfPolygon ( Span< const Length2 > vertices )

Gets the area of a polygon.

Note: This function is valid for any non-self-intersecting (simple) polygon, which can be convex or concave.; Winding order doesn't matter.

◆ GetDelta()

Angle playrho::GetDelta	(	Angle	a1,
		Angle	a2
	)

noexcept

Gets the shortest angular distance to go from angle 1 to angle 2.

This gets the angle to rotate angle 1 by in order to get to angle 2 with the least amount of rotation.

Returns: Angle between -Pi and Pi radians inclusively.

See also: GetNormalized

◆ GetFwdPerpendicular()

template<class T >

constexpr auto playrho::GetFwdPerpendicular ( const T vector )

constexprnoexcept

Gets a vector clockwise (forward-clockwise) perpendicular to the given vector.

This takes a vector of form (x, y) and returns the vector (y, -x).

Parameters

vector Vector to return a clockwise perpendicular equivalent for.

Returns: A clockwise 90-degree rotation of the given vector.

See also: GetRevPerpendicular.

Referenced by playrho::d2::ConvexHull::Get().

◆ GetInverse22()

constexpr Mat33 playrho::GetInverse22 ( const Mat33 & value )

constexprnoexcept

Gets the inverse of the given matrix as a 2-by-2.

Returns: Zero matrix if singular.

Referenced by playrho::d2::InitVelocity().

◆ GetMagnitude()

template<typename T >

auto playrho::GetMagnitude ( T value )

inline

Gets the magnitude of the given value.

Note: Works for any type for which GetMagnitudeSquared also works.

Examples: DistanceJoint.cpp, PulleyJoint.cpp, and World.cpp.

Referenced by playrho::d2::Simplex::CalcMetric(), playrho::d2::GetCentripetalForce(), playrho::d2::GetCurrentLengthA(), playrho::d2::GetCurrentLengthB(), playrho::d2::GetDistanceJointConf(), playrho::d2::ChainShapeConf::GetMassData(), playrho::d2::GetMassData(), playrho::d2::GetPulleyJointConf(), playrho::Normalize(), and playrho::d2::SolvePosition().

◆ GetMagnitudeSquared()

template<typename T >

constexpr auto playrho::GetMagnitudeSquared ( T value )

constexprnoexcept

Gets the square of the magnitude of the given iterable value.

Note: For performance, use this instead of GetMagnitude(T value) (if possible).

Returns: Non-negative value from 0 to infinity, or NaN.

Referenced by playrho::d2::CalcGravitationalAcceleration(), playrho::d2::Cap(), playrho::d2::Distance(), playrho::d2::VertexSet::find(), playrho::d2::FindClosestBody(), playrho::d2::GetConvexHullAsVector(), playrho::d2::GetLocalRotInertia(), playrho::GetMagnitude(), playrho::d2::GetManifold(), playrho::d2::GetMassData(), playrho::d2::GetToiViaSat(), playrho::d2::IsUnderActive(), playrho::d2::RayCast(), playrho::d2::Body::SetAcceleration(), playrho::d2::SolvePosition(), playrho::d2::SolveVelocity(), playrho::d2::TestOverlap(), playrho::d2::TestPoint(), and playrho::d2::WorldImpl::Update().

◆ GetNormalized()

Angle playrho::GetNormalized ( Angle value )

inlinenoexcept

Gets the "normalized" value of the given angle.

Returns: Angle between -Pi and Pi radians inclusively where 0 represents the positive X-axis.

See also: Atan2

Referenced by playrho::GetDelta(), and playrho::d2::GetNormalized().

◆ GetPolarMoment()

SecondMomentOfArea playrho::GetPolarMoment ( Span< const Length2 > vertices )

Gets the polar moment of the area enclosed by the given vertices.

Warning: Behavior is undefined if given collection has less than 3 vertices.

Parameters

vertices Collection of three or more vertices.

Referenced by playrho::d2::GetMassData().

◆ GetRevPerpendicular()

template<class T >

constexpr auto playrho::GetRevPerpendicular ( const T vector )

constexprnoexcept

Gets a vector counter-clockwise (reverse-clockwise) perpendicular to the given vector.

This takes a vector of form (x, y) and returns the vector (-y, x).

Parameters

vector Vector to return a counter-clockwise perpendicular equivalent for.

Returns: A counter-clockwise 90-degree rotation of the given vector.

See also: GetFwdPerpendicular.

◆ GetRevRotationalAngle()

constexpr Angle playrho::GetRevRotationalAngle	(	Angle	a1,
		Angle	a2
	)

constexprnoexcept

Gets the reverse (counter) clockwise rotational angle to go from angle 1 to angle 2.

Returns: Angular rotation in the counter clockwise direction to go from angle 1 to angle 2.

◆ GetSymInverse33()

constexpr Mat33 playrho::GetSymInverse33 ( const Mat33 & value )

constexprnoexcept

Gets the symmetric inverse of this matrix as a 3-by-3.

Returns: Zero matrix if singular.

Referenced by playrho::d2::InitVelocity().

◆ IsOdd()

template<typename T >

constexpr bool playrho::IsOdd ( T val )

constexprnoexcept

Is-odd.

Determines whether the given integral value is odd (as opposed to being even).

Referenced by playrho::d2::GetToiViaSat().

◆ ModuloViaFmod()

template<typename T >

auto playrho::ModuloViaFmod	(	T	dividend,
		T	divisor
	)

inlinenoexcept

Modulo operation using std::fmod.

Note: Modulo via std::fmod appears slower than via std::trunc.

See also: ModuloViaTrunc

Referenced by playrho::GetNormalized().

◆ ModuloViaTrunc()

template<typename T >

auto playrho::ModuloViaTrunc	(	T	dividend,
		T	divisor
	)

inlinenoexcept

Modulo operation using std::trunc.

Note: Modulo via std::fmod appears slower than via std::trunc.

See also: ModuloViaFmod

Referenced by playrho::GetNormalized().

◆ NextPowerOfTwo()

template<typename T >

constexpr T playrho::NextPowerOfTwo ( T x )

constexpr

Gets the next largest power of 2.

Given a binary integer value x, the next largest power of 2 can be computed by a S.W.A.R. algorithm that recursively "folds" the upper bits into the lower bits. This process yields a bit vector with the same most significant 1 as x, but all one's below it. Adding 1 to that value yields the next largest power of 2.

Referenced by playrho::d2::DynamicTree::Reserve().

◆ Secant()

template<typename T , typename U >

constexpr U playrho::Secant	(	T	target,
		U	a1,
		T	s1,
		U	a2,
		T	s2
	)

constexprnoexcept

Secant method.

See also: https://en.wikipedia.org/wiki/Secant_method

Referenced by playrho::d2::GetToiViaSat().

◆ Solve()

template<typename T , typename U >

constexpr auto playrho::Solve	(	const Matrix22< U >	mat,
		const Vector2< T >	b
	)

constexprnoexcept

Solves A * x = b, where b is a column vector.

Note: This is more efficient than computing the inverse in one-shot cases.

Referenced by playrho::d2::SolvePosition().

◆ Solve22()

template<typename T >

constexpr T playrho::Solve22	(	const Mat33 &	mat,
		const T	b
	)

constexprnoexcept

Solves A * x = b, where b is a column vector.

Note: This is more efficient than computing the inverse in one-shot cases.; Solves only the upper 2-by-2 matrix equation.

Referenced by playrho::d2::SolvePosition(), and playrho::d2::SolveVelocity().

◆ Solve33()

constexpr Vec3 playrho::Solve33	(	const Mat33 &	mat,
		const Vec3	b
	)

constexprnoexcept

Solves A * x = b, where b is a column vector.

Note: This is more efficient than computing the inverse in one-shot cases.

Referenced by playrho::d2::SolvePosition(), and playrho::d2::SolveVelocity().

◆ Transform()

template<std::size_t M, typename T1 , std::size_t N, typename T2 >

constexpr auto playrho::Transform	(	const Vector< T1, M >	v,
		const Matrix< T2, M, N > &	m
	)

constexprnoexcept

Multiplies an M-element vector by an M-by-N matrix.

Parameters

v	Vector that's interpreted as a matrix with 1 row and M-columns.
m	An M-row by N-column transformation matrix to multiply the vector by.

See also: https://en.wikipedia.org/wiki/Transformation_matrix

Functions

Detailed Description

Function Documentation

◆ AlmostZero()

◆ Atan2()

◆ Bisect()

◆ ComputeCentroid()

◆ Cross()

◆ Dot()

◆ GetAngle()

◆ GetAreaOfPolygon()

◆ GetDelta()

◆ GetFwdPerpendicular()

◆ GetInverse22()

◆ GetMagnitude()

◆ GetMagnitudeSquared()

◆ GetNormalized()

◆ GetPolarMoment()

◆ GetRevPerpendicular()

◆ GetRevRotationalAngle()

◆ GetSymInverse33()

◆ IsOdd()

◆ ModuloViaFmod()

◆ ModuloViaTrunc()

◆ NextPowerOfTwo()

◆ Secant()

◆ Solve()

◆ Solve22()

◆ Solve33()

◆ Transform()